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STABILITY AND HOPF BIFURCATIONS IN A DELAYED PREDATOR–PREY SYSTEM WITH A DISTRIBUTED DELAY

CUN-HUA ZHANG

Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P. R. China

Author for correspondence.

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XIANG-PING YAN

Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, P. R. China

Supported by "QingLan" Talent Engineering of Lanzhou Jiaotong University (QL-05-20A).

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https://doi.org/10.1142/S0218127409024062Cited by:4 (Source: Crossref)

Abstract

This paper is concerned with a delayed Lotka–Volterra two-species predator–prey system with a distributed delay. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that the positive equilibrium of the system is always locally asymptotically stable when the delay kernel is the weak kernel while there is a stability switch of positive equilibrium when the delay kernel is the strong kernel and the system can undergo a Hopf bifurcation at the positive equilibrium when the average time delay in the delay kernel crosses certain critical values. In particular, by applying the normal form theory and center manifold reduction to functional differential equations (FDEs), the explicit formula determining the direction of Hopf bifurcations and the stability of bifurcated periodic solutions is given. Finally, some numerical simulations are also included to support the analytical results obtained.

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